Do the pdf and the pmf and the cdf contain the same information?
For me the pdf gives the whole probability to a certain point(basically the area under the probability).
The pmf give the probability of a certain point.
The cdf give the probability under a certain point.
So to me the pdf and cdf have the same information, but the pmf does not because it gives the probability for a point x on the distribution.
Best Answer
Where a distinction is made between probability function and density*, the pmf applies only to discrete random variables, while the pdf applies to continuous random variables.
* formal approaches can encompass both and use a single term for them
The cdf applies to any random variables, including ones that have neither a pdf nor pmf
(such as a mixed distribution - for example, consider the amount of rain in a day, or the amount of money paid in claims on a property insurance policy, either of which might be modelled by a zero-inflated continuous distribution).
The cdf for a random variable $X$ gives $P(X\leq x)$
The pmf for a discrete random variable $X$, gives $P(X=x)$.
The pdf doesn't itself give probabilities, but relative probabilities; continuous distributions don't have point probabilities. To get probabilities from pdfs you need to integrate over some interval - or take a difference of two cdf values.
It's difficult to answer the question 'do they contain the same information' because it depends on what you mean. You can go from pdf to cdf (via integration), and from pmf to cdf (via summation), and from cdf to pdf (via differentiation) and from cdf to pmf (via differencing), so when you have a pmf or a pdf, it contains the same information as the cdf.